HINT: <no title>
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Look at the points given: are they at the peaks (maximum
values) of the cosine curve or at the troughs (minimum values)? Or are
they in the 'middle' of the graph? Use these points to decide how tall
the graph is: from that you can find the amplitude of the graph,
a.
STEP: Determine the amplitude of the curve
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There are some important points on a cosine graph. These are the
maximum points, the minimum points, and the points in the middle.
With those points we can figure out the values of a and q.
The graph below shows the maximum, minimum and middle values of
the equation in this question. The sinusoidal axis,
which is the line in the middle, sits halfway from the top to the bottom.
In other words, it is exactly halfway between the maximum and minimum!
The picture also shows the amplitude of the graph. It is the distance
from the middle to the maximum. Note that the amplitude is
always positive because it is a distance.
In this question, we have one point at the bottom and another point at the top.
Let's start by finding the amplitude of the graph, which is the
value of a in the equation. The y-value at
the bottom is
−2,5,
while the y-value at the the top is
0,5.
The distance from the top to the bottom is 0,5−(−2,5)=3. The amplitude is half of that. So the amplitude of this graph, and the value of a, is 32.
At this point, we know that the equation is y=32cosx+q.
STEP: Determine the vertical shift of the curve
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Now on to finding the value of q. This can actually be done very quickly. Normally the sinusiodal axis sits at y=0. But the q term moves that axis up or down. In this graph, the sinusoidal axis is at y=−1; the graph has been shifted down 1 space. Therefore q=−1.
The equation for the graph shown is y=32cosx−1.
The correct answers are a=32 and q=−1.
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